SEMI-LINEAR LINEAR INDEX MODEL WHEN THE LINEAR COVARIATES AND INDICES ARE INDEPENDENT

Summary Most dimension reduction models are suited for continuous but not for discrete covariates. A flexible model to in-corporate both discrete and continuous covariates is to assume that some covariates, a q-dimensional vector Z, are related to the response variable, Y, through a linear relationship; while the remaining covari-ates, a p-dimensional vector X, are related to Y through k indices which areX ′B and some unknown function g. This results in a semi-parametric model called semi-linear index model, which includes both the popular single-index model and the partial linear model as special cases. To avoid the curse of dimensionality, k should be much smaller than p, and this is often realistic as the key features of a high di-mensional variable can often be extracted through a low-dimensional subspace. Two approaches to estimate the model components have been considered by Carroll, Fan, Gijbels and Wand (1997) and Yu and Ruppert (2002). Both focus on the simple case where k = 1, a